SU(2) contractions

[Aieletf]

Hello,

I want to implement a Lagrangian term of the following form:

$ X^'dagger_i \tau^{n,a}_{ij} X_j H^'dagger_k \tau^{2,a}_{kl} H_l $

Where $tau^{n/2}$ are the SU(2) generators at the n-tuplet / doublet representation, and X is an n vector of SU(2). $a=1,2,3$ is an SU(2) index.
I've been trying to use the THDM-triplet example but I fail to understand the indices of, for example, the following form:

$ K1 Delta[lef1, lef2] Delta[lef1b, lef4b] Delta[lef2b, lef3b] Delta[lef3,lef4] conj[T].T.conj[T].T $

Is there an SU(2) invariant documentation of the triplet Lagrangian that I can use?

Thanks,
Aielet

[FStaub]

Hi,

maybe this description is useful for you:
http://stauby.de/sarah_wiki/index.php?t ... ntractions

Note, you need to write the fields charged under the broken SU(2) as tensors product of fundamental irreps, not vectors: a 3 is a 2x2 matrix, a 4 is a tensor with 3 indices, a 7 has 6 indices,...

Cheers,
Florian

[Guest]

To get the feeling for the index structure, I'm starting with an SU(2) doublet example.

I have the following fields:
ScalarFields[[1]] = {H1, 1, {H1p, H10}, 1/2, 2, 1};
ScalarFields[[2]] = {H2, 1, {H2p, H20}, 1/2, 2, 3};

and I want to write explicitly the indices for the following interaction:

conj[H2].H2.conj[H1].H1

I have tried several options which are all suppose to be equivalent, but each is giving me different results for the quartic RGE's.
I think I'm miss-interpreting the notations. Can you please tell me the full index structure of this Lagrangian term?

Thanks
Aielet

[FStaub]

Hi,

What did you try? I would say the correct one is

Code: Select all

Delta[lef1,lef2] Delta[col1,col2] Delta[lef3,lef4]

Cheers
Florian

[Guest]

This is one of the options I tried. My problem is the following. When I'm omitting the indices, i.e. I have in my Lagrangian

Lambda3 conj[H2].H2.conj[H1].H1

I'm getting the following one-loop RGE for Lambda3

{Lambda3, 2 Lambda3 (3 Lambda1 + 7 Lambda2 + 2 Lambda3)}

which is the correct result. But, when I'm specifying the indices as you wrote:

Lambda3 Delta[lef1,lef2] Delta[col1,col2] Delta[lef3,lef4] conj[H2].H2.conj[H1].H1

I'm getting

{Lambda3, 2 Lambda3 (3 Lambda1 + 7 Lambda2 - 2 Lambda3)}

Note the different signs in the last term.

I'll appreciate your help.
Thanks
Aielet

[FStaub]

That's strange. Can you compare the content of

Code: Select all

SA`SSSSlist

after initializing the model?

[Guest]

Sure (sorry for the delay)

without specifying the contractions I get:

{{{conj[H1[{lef1}][{gen1}]], conj[H1[{lef3}][{gen3}]],
H1[{lef2}][{gen2}], H1[{lef4}][{gen4}]},
1/2 Lambda1 Delta[lef1, lef2] Delta[lef3, lef4],
False}, {{conj[H1[{lef3}][{gen3}]], conj[H2[{lef1}][{gen1, col1}]],
H1[{lef4}][{gen4}], H2[{lef2}][{gen2, col2}]},
Lambda3 Delta[col1, col2] Delta[lef1, lef2] Delta[lef3, lef4],
False}, {{conj[H2[{lef1}][{gen1, col1}]],
conj[H2[{lef3}][{gen3, col3}]], H2[{lef2}][{gen2, col2}],
H2[{lef4}][{gen4, col4}]},
1/2 Lambda2 Delta[col1, col2] Delta[col3, col4] Delta[lef1,
lef2] Delta[lef3, lef4], False}}

and the second option gives:

{{{conj[H1[{lef1}][{gen1}]], conj[H1[{lef3}][{gen3}]],
H1[{lef2}][{gen2}], H1[{lef4}][{gen4}]},
1/2 Lambda1 Delta[lef1, lef2] Delta[lef3, lef4],
False}, {{conj[H1[{lef3}][{gen3}]], conj[H2[{lef1}][{gen1, col1}]],
H1[{lef4}][{gen4}], H2[{lef2}][{gen2, col2}]},
Lambda3 Delta[col1, col2] InvMat[4][lef1, lef2, lef3, lef4],
False}, {{conj[H2[{lef1}][{gen1, col1}]],
conj[H2[{lef3}][{gen3, col3}]], H2[{lef2}][{gen2, col2}],
H2[{lef4}][{gen4, col4}]},
1/2 Lambda2 Delta[col1, col2] Delta[col3, col4] Delta[lef1,
lef2] Delta[lef3, lef4], False}}

I also tried to contract the fields with:

Delta[col1, col2] Delta[lef1, lef2] Delta[lef3, lef4]

and I still get Delta[col1, col2] InvMat[4][lef1, lef2, lef3, lef4] as the contraction in SA`SSSSlist.

[FStaub]

Looks strange. Sorry, but I'm right now changing countries, i.e. it's not possible for me to take a close look. I'll come back to you asap.

Cheers,
Florian